RSystemsCategoryOfM 2009-01-15 21:05:03 2009-01-26 01:40:22 Topic MR-system general $(M R)$--system $(M R)$--systems category of $(M R)$--systems category of automata relational biology % this is the default PlanetPhysics preamble. as your knowledge % there are many more packages, add them here as you need them % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@ }}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \subsection{Introduction and the basic concept of a metabolic-replication system} Robert Rosen introduced \emph{metabolic--repair models}, or $(M,R)$-systems in mathematical biology (\emph{abstract relational biology}) in 1957 (\cite{RRosen1, RRosen2}); such systems will be here abbreviated as $MR$-systems, (or simply $MR$'s). Rosen, then represented the $MR$'s in terms of categories of sets, deliberately selected without any structure other than the {\em discrete topology of sets}. \begin{definition} The simplest $MR$-system represents a relational model of the primordial organism which is defined by the following \emph{categorical sequence (or diagram) of sets and set-theoretical mappings}: $f: A \rightarrow B, \phi: B \rightarrow Hom_{MR}(A,B)$, where $A$ is the set of inputs to the $MR$-system, $B$ is the set of its outputs, and $\phi$ is the `repair map', or $R$-component, of the $MR$-system which associates to a certain product, or output $b$, the `metabolic' component (such as an enzyme, E, for example) represented by the set-theoretical mapping $f$. Then, $Hom_{MR}(A,B)$ is defined as the set of all such metabolic (set-theoretical) mappings (occasionally written incorrectly by some authors as $\left\{f\right\}$). \end{definition} \begin{definition} A \emph{general $(M,R)$-system} was defined by Rosen (1958a,b) as the network or graph of the metabolic and repair components that were specified above in \textbf{Definition 0.1}; such components are networked in a complex, abstract `organism' defined by all the abstract relations and connecting maps between the sets specifying all the metabolic and repair components of such a general, abstract model of the biological organism. The mappings bettwen $(M,R)$-systems are defined as the the metabolic and repair set-theoretical mappings, such as $f$ and $\phi$ (specified in \textbf{Definition 0.1}); moreover, there is also a finite number of sets (just like those that are defined as in \textbf{Definition 0.1}): $A_i, B_i$, whereas $f \in Hom_{MR_i}(A_i,B_i)$ and $\phi \in Hom_{MR_i}[B, Hom_{MR_i}(A_i,B_i)]$, with $i \in I$, and $I$ being a finite index set, or directed set, with $(f,\phi)$ being a finite number of distinct metabolic and repair components pairs. Alternatively, one may think of a a general $MR$-system as being `made of' a finite number $N$ of interconnected $MR_i$, metabolic-repair modules with input sets $A_i$ and output sets $B_i$. To sum up: a \emph{general MR-system} can be defined as a \emph{family of interconnected quartets}: $\left\{(A_i, B_i, f_i, \phi_i)\right\}_{i \in I}$, where $I$ is an index set of integers $i=1, 2, ..., n$. \end{definition} \subsection{Category of (M,R)--systems} \begin{definition} A \emph{category of $(M,R)$-system quartet modules}, $\left\{(A_i, B_i, f_i, \phi_i)\right\}_{i \in I}$, with I being an index set of integers $i=1,2,..., n$, is a small category of sets with set-theoretical mappings defined by the MR-morphisms between the quarted modules $\left\{(A_i, B_i, f_i, \phi_i)\right\}_{i \in I}$, and also with repair components defined as $\phi_i \in Hom_{MR_i}[B, Hom_{MR_i}(A_i,B_i)]$, with the $(M,R)$-morphism composition defined by the usual composition of functions between sets. With a few, additional notational changes it can be shown that the category of $(M,R)$-systems is a subcategory of the category of automata (or sequential machines), $\mathcal{S}_{[M,A]}$ (\cite{ICB73, ICBM74}). \end{definition} \begin{remark} For over two decades, Robert Rosen developed with several coworkers the MR-systems theory and its applications to life sciences, medicine and general systems theory. He also considered biocomplexity to be an `emergent', defining feature of organisms which is \emph{not reducible in terms of the molecular structures} (or molecular components) of the organism and their physicochemical interactions. However, in his last written book in 1997 on \emph{``Essays on Life Itself", published posthumously in 2000}, Robert Rosen finally accepted the need for representing organisms in terms of {\em categories with structure} that entail biological functions, both metabolic and repair ones. Note also that, unlike Rashevsky in his theory of organismic sets, Rosen did not attempt to extend the $MR$s to modeling societies, even though with appropriate modifications of \emph{generalized $(M,R)$-system categories with structure} (\cite{ICB73, ICBM74, ICB87a}), this is feasible and yields meaningful mathematical and sociological results. Thus, subsequent publications have generalized MR-system (GMRs) and have studied the fundamental, mathematical properties of algebraic categories of GMRs that were constructed functorially based on the {Yoneda-Grothendieck Lemma} and construction. Then it was shown that such algebraic categories of GMRs are \emph{Cartesian closed} \cite{ICB73}. Several \emph{molecular biology realizations of GMRs} in terms of DNA, RNAs, enzymes, $RNA \to DNA$-reverse trancriptases, and other biomolecular components were subsequently introduced and discussed in ref. \cite{BBGGk6,ICB87a, ICB87b} in terms of \emph{non-linear genetic network models} in many-valued, $LM_n$ logic algebras (or \PMlinkname{algebraic category $\mathcal{LM}$ of $LM_n$ logic algebras}{AlgebraicCategoryOfLMnLogicAlgebras}). \end{remark} \begin{thebibliography}{9} \bibitem{Rashevsky1-yr1965} Rashevsky, N.: 1965, The Representation of Organisms in Terms of Predicates, \emph{Bulletin of Mathematical Biophysics} \textbf{27}: 477-491. \bibitem{Rashevsky2-1969} Rashevsky, N.: 1969, Outline of a Unified Approach to Physics, Biology and Sociology., \emph{Bulletin of Mathematical Biophysics} \textbf{31}: 159--198. \bibitem{Rosenbook} Rosen, R.: 1985, \emph{Anticipatory Systems}, Pergamon Press: New York. \bibitem{RRosen1} Rosen, R.: 1958a, A Relational Theory of Biological Systems \emph{Bulletin of Mathematical Biophysics} \textbf{20}: 245-260. \bibitem{RRosen2} Rosen, R.: 1958b, The Representation of Biological Systems from the Standpoint of the Theory of Categories., \emph{ Bulletin of Mathematical Biophysics} \textbf{20}: 317-341. \bibitem{RRosen3} Rosen, R.: 1987, On Complex Systems, \emph{European Journal of Operational Research} \textbf{30}:129--134. \bibitem{ICB73} Baianu, I.C.: 1973, Some Algebraic Properties of \emph{\textbf{(M,R)}} -- Systems. \emph{Bulletin of Mathematical Biophysics} \textbf{35}, 213-217. \bibitem{ICBM74} Baianu, I.C. and M. Marinescu: 1974, On A Functorial Construction of \emph{\textbf{(M,R)}}-- Systems. \emph{Revue Roumaine de Mathematiques Pures et Appliqu\'ees} \textbf{19}: 388-391. \bibitem{ICB80} Baianu, I.C.: 1980, Natural Transformations of Organismic Structures., \emph{Bulletin of Mathematical Biology},\textbf{42}: 431-446. \bibitem{ICB77} I.C. Baianu: 1977, A Logical Model of Genetic Activities in \L{}ukasiewicz Algebras: The Non-linear Theory. \emph{Bulletin of Mathematical Biophysics}, \textbf{39}: 249-258. \bibitem{ICB8} I.C. Baianu: 1983, Natural Transformation Models in Molecular Biology., in \emph{Proceedings of the SIAM Natl. Meet}., Denver, CO.; \PMlinkexternal{An Eprint is here available}{http://cogprints.org/3675/1/Naturaltransfmolbionu6.pdf} . \bibitem{ICB9} I.C. Baianu: 1984, A Molecular-Set-Variable Model of Structural and Regulatory Activities in Metabolic and Genetic Networks., \emph{FASEB Proceedings} \textbf{43}, 917. \bibitem{ICB87a} I.C. Baianu: 1987a, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.), \emph{Mathematical Models in Medicine}, vol. 7., Pergamon Press, New York, 1513--1577; \PMlinkexternal{CERN Preprint No. EXT-2004-072:}{http://documents.cern.ch/cgi-bin/setlink?base=preprint&categ=ext&id=ext-2004-067}. \bibitem{ICB87b} I.C. Baianu: 1987b, Molecular Models of Genetic and Organismic Structures, in \emph{Proceed. Relational Biology Symp.} Argentina; \PMlinkexternal{CERN Preprint No.EXT-2004-067:MolecularModelsICB3.doc}{http://documents.cern.ch/cgi-bin/setlink?base=preprint&categ=ext&id=ext-2004-067}. \bibitem{Bgg2} I.C. Baianu, Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued \L ukasiewicz Algebras in Relation to Dynamic Bionetworks, \textbf{(M,R)}--Systems and Their Higher Dimensional Algebra, \PMlinkexternal{Abstract of Report is here available as a PDF}{http://www.ag.uiuc.edu/fs401/QAuto.pdf} and \PMlinkexternal{html document}{http://doc.cern.ch/archive/electronic/other/ext/ext-2004-058/QuantumAutnu3_ICB.pdf} \bibitem{BGB2} R. Brown, J. F. Glazebrook and I. C. Baianu: A categorical and higher dimensional algebra framework for complex systems and spacetime structures, \emph{Axiomathes} \textbf{17}:409--493. (2007). \bibitem{LO68} L. L$\ddot{o}$fgren: 1968. On Axiomatic Explanation of Complete Self--Reproduction. \emph{Bull. Math. Biophysics}, \textbf{30}: 317--348. \bibitem{ICB2004a} Baianu, I.C.: 2004a. \L{}ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint. Cogprints--Sussex Univ. \bibitem{ICB04b} Baianu, I.C.: 2004b \L{}ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. \textit{Health Physics and Radiation Effects} (June 29, 2004). \bibitem{ICB2k6} Baianu, I. C.: 2006, Robert Rosen's Work and Complex Systems Biology, \emph{Axiomathes} \textbf{16}(1--2):25--34. \bibitem{BBGGk6} Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and \L{}ukasiewicz--Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., \emph{Axiomathes}, \textbf{16} Nos. 1--2: 65--122. \end{thebibliography}